Secure CDMA and Frequency Hop Sequences

Single sequences like Legendre have high linear complexity. Known CDMA families of sequences all have low linear complexities. A new method is presented for constructing CDMA sequence sets with the linear complexity of the Legendre sequences, from new frequency hop patterns. Linear complexity is a measure of the robustness of a sequence to cryptographic attack. These are the first families whose normalized linear complexities do not asymptote to 0, verified for lengths up to 6×108. The new constructions in array format are also useful in watermarking images, where the high linear complexity prevents an attacker from reconstructing the watermark from any part of it. We present a conjecture regarding the recursion polynomials. We also have a method to reverse the process, and from small Kasami/No-Kumar sequences we obtain a new family of 2n doubly periodic (2n+1)×(2n-1) frequency hop patterns with correlation 2.